Resonance and Antiresonance Frequencies

An excellent reference describing appropriate ways of measuring the piezoelectric coefficients of bulk materials is the IEEE Standard for Piezoelectricity [1], In brief, the method entails choosing a sample with a geometry such that the desired resonance mode can be excited, and there is little overlap between modes. Then, the sample is electrically excited with an alternating field, and the impedance (or admittance, etc.) is measured as a function of frequency. Extrema in the electrical responses are observed near the resonance and antiresonance frequencies. As an example, consider the length extensional mode of a vibrator. Here the elastic compliance under constant field can be measured from... 

Alternatively, using Eq. (6.62), kn may be determined directly from the resonance and antiresonance frequencies of a bar poled across its thickness and using Eq. (6.62)... 

The portion of the electrical circuit representation of a system that delivers the oscillatory resonance behavior at one or more rescmance frequencies ( resonance and antiresonance frequencies). 

The resonance and antiresonance frequencies are associated with the frequencies of minimum and maximum impedance for a particular resonance phenomenon, respectively. A continuous structure, like a cantilever beam, may have many resonance and antiresonance frequencies alternating in an ordered fashion with respect to frequency. 

The longitudinal resonance (the lengthwise poled rod supported at its centre (node)) and antiresonance frequencies are 71.8 kHz and 88.1 kHz. 

Figure 4.1.9. In the resonance state, large strain amplitudes and large capacitance changes (called motional capacitance) are induced, and the current can easily flow into the device. On the other hand, at antiresonance, the strain induced in the device compensates completely, resulting in no capacitance change, and the current cannot flow easily into the sample. Thus, for a high k material the first antiresonance frequency/a should be twice as large as the first resonance frequency/r.

In a typical case, where ksi = 0.3, the antiresonance state varies from the previously mentioned mode and becomes closer to the resonance mode. The low-coupling material exhibits an antiresonance mode where capacitance change due to the size change is compensated completely by the current required to charge up the static capacitance (called damped capacitance). Thus, the antiresonance frequency/a will approach the resonance frequency/r. 

The material requirements for these classes of devices are somewhat different, and certain compounds will be better suited to particular applications. The ultrasonic motor, for instance, requires a very hard piezoelectric with a high mechanical quality factor Qm, to suppress heat generation. Driving the motor at the antiresonance frequency, rather than at resonance, is also an intriguing technique to reduce the load on the piezoceramic and the power supply [42]. The servo displacement transducer suffers most from strain hysteresis and, therefore, a PMN electrostrictor is used for this purpose. The pulse drive motor requires a low permittivity material aimed at quick response with a certain power supply rather than a small hysteresis, so soft PZT piezoelectrics are preferred rather than the high-permittivity PMN for this application. 

The crystal is normally operated at frequencies located between the series resonance and the parallel or so-called antiresonance frequencies. This defines a useful bandwidth for the crystal relative to the series... 

Values of the coupling factors are determined by the piezoelectric resonance method. This analyses the impedance Z of the material when it is excited by a voltage source and uses the resonant frequency, at which Z = 0, and the antiresonant frequency, at which Z is infinite, to calculate the appropriate k [32]. This method is satisfactory for ceramics, but large errors occur when it is applied directly to the polymers, because of their large mechanical losses and subsequent high damping effects. It is necessary to use equivalent-circuit models to select the parameters that produce resonance curves that are best fitted to observed ones [33]. 

This describes the filter coefficients in terms of an exponential damping parameter (r for the zeroes, for the poles) and a center frequency of resonance (antiresonance for the zeroes), which is Freq for the zeroes and Freq for the poles. We can now control aspects of the filter more directly from these parameters, knowing that once we decide on r and Freq, we can convert to a, a, b, and directly. Figure 3.11 shows the BiQuad in block diagram form. 

Further refinement can be achieved by the use of zeros. These can be used to create antiresonances, corresponding to a notch in the frequency response. Here the format synthesis model again deviates from the all-pole tube model, but recall that we only adopted the all-pole model to make the derivation of the tube model easier. While the all-pole model has been shown to be perfectly adequate for vowel sounds, the quality of nasal and fricative sounds can be improved by the use of some additional zeros. In particular, it has be shown [254] that the use of a single zero anti-resonator in series with a the normal resonators can produce realistic nasal sounds. 

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